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WASHINGTON — Driving across the flatlands of Illinois with Barack Obama during the Senate race of 2004, Kevin Thompson sometimes found himself tutoring the candidate on gay rights.

Thompson, then a traveling aide, recalls long conversations about topics like the 1969 Stonewall Rebellion that sparked the gay rights movement, gay adoption — Obama once volunteered that Thompson and his partner would make “great parents,’’ Thompson recalled — and same-sex marriage, which Obama has in the past opposed.

Thompson, an Obama supporter, is skeptical about that. “To this day,’’ he said, “I don’t think Barack Obama has any issue with two people of the same gender getting married.’’

Now Obama says his views on same-sex marriage are evolving, and as he runs for reelection he is seeking support from gay donors who want to know where he stands.

This week, he will headline a $1,250-a-plate “Gala with the Gay Community’’ in New York, his first such event as president; on June 29, he will host a Gay Pride reception at the White House. He is doing so at a time when the New York Legislature is considering whether to make same-sex marriage legal — a vote that the president will no doubt be asked about while in New York.

The White House would not comment on whether Obama was ready to endorse same-sex marriage.

But one Democratic strategist close to the White House, speaking on the condition of anonymity, said some senior advisers “are looking at the tactics of how this might be done if the president chose to do it.’’

And Representative Barney Frank, a Massachusetts Democrat who is gay, said in an interview that a top adviser to Obama, whom he would not name, asked him this year, “What would be the effect if he came out for same-sex marriage?’’

“My own view is that I look at President Obama’s record, he was probably inclined to think that same-sex marriage was legitimate, but as a candidate for president in 2008 that would have been an unwise thing to say,’’ Frank said. “And I don’t mean that he’s being hypocritical. I mean that if you live in a democratic society, it is a mix of what you think the voters want and what you think is doable.’’

Many gay leaders say because the president has a strong record on issues they care about — prodding Congress to repeal the “don’t ask, don’t tell’’ policy, which barred openly gay men and lesbians from serving in the military, and withdrawing legal support for the Defense of Marriage Act, which defines marriage as between a man and a woman — he is not under pressure to announce a change in his position before the 2012 election.

But with the political climate around gay rights changing drastically — a handful of recent polls show that Americans, by a slim majority, now support same-sex marriage — some strategists see little political cost to a shift in position.

And a review of Obama’s record, dating to when he first ran for public office, suggests that he may have been for same-sex marriage before he was against it.

In 1996, as a candidate for the State Senate in Illinois, Obama responded to a questionnaire from a gay newspaper. “I favor legalizing same-sex marriages,’’ Obama wrote, “and would fight efforts to prohibit such marriages.’’

White House officials have said Obama was really referring to civil unions, which he supports. (On Friday, Obama’s communications director, Dan Pfieffer, caused a brief kerfuffle by telling a conference of bloggers that Obama had not filled out the forms himself; the White House later said he was mistaken.)

By the time Obama ran for the US Senate in 2004, his position had become more nuanced.

Jackie Kaplan, a Chicago Democrat who was cochairwoman of a committee of gays and lesbians supporting Obama, said he raised practical objections and made the case this way: “Why spend a lot of time on an issue that is not going to happen? The Defense of Marriage law is on the books, we’re not going to overturn that, let’s talk about how we can build more equality.’’

Tracy Baim, a gay journalist in Chicago who interviewed Obama in 2004, remembers the candidate asking her to turn off her tape recorder so they could have a candid conversation on same-sex marriage.

She said his objections were based on what he saw as realistic considerations: “I know what you want, I know what you can get.’’

But when his Senate campaign moved into the general election against Alan Keyes, Obama told an interviewer for a black-owned radio station that religion was a factor.

Kaplan said she felt that Obama was “pandering to Alan Keyes’’ or setting himself up to run for higher office; Baim, who said Obama had not cited his religious beliefs to her, viewed it as “a political maneuver.’’